Guest post by Colin Beveridge from http://www.flyingcoloursmaths.co.uk/
“The race is not always to the swift, nor the fight to the strong, but that’s the way to bet.” – Damon Runyan
Alice* is a professional gambler. Over the last two years she has consistently churned out around £2,000 profit per month by finding value bets on a handful of different sports and a variety of odds at Betfair, and the dozen bookies where she has accounts. (Because she wouldn’t reveal her real results to me, I’ve simulated them on the computer.)
Bob*, on the other hand, is a billionaire whose only bet over the last two years was to put £100,000 on Man United to beat Wolves at 1/4, which they duly did.
Here’s a summary of how they’ve performed in 24 months
By any of the measures in the table, it seems like Bob is better at gambling than Alice is. He did less work, had a 100% record, staked less, won more and got a better return on his investment. And yet, that doesn’t seem to stack up: Bob made one winning prediction; Alice consistently wins more than she loses – even if she is wrong more often than she is right.
So, how can we tell the difference between winning because you’re an insightful genius and winning because you’re a lucky millionaire? Well… we can’t. All we can do, is to say how likely you are to have won by luck or skill.
Let’s assume that the odds offered by the bookies are fair. This is a slightly harsh assumption on Alice and Bob – in reality the bookies do their best to have a slight edge and it’s easier to break even with fair odds than with odds biased against you. So this assumption underestimates how skilful Alice and Bob really are.
It’s easy to figure out how lucky Bob was. Assuming the odds are fair, 1/4 (or 1.25) translates to an 80% chance – so the probability of Bob doing as well as he did by luck alone is 80%. His skill accounts for the other 20%.
It’s much tougher to calculate how lucky Alice was.
If Alice made 1600 bets at fair odds she would expect to break even… give or take. It’s the give or take that determines how lucky or skilful she is. If she makes more than the ‘give or take’ margin she’d expect then she’s more likely to be skilful than lucky. If her profits are within the margin, we’d think of her as lucky. The first tricky bit is figuring out the ‘give or take’.
Things are complicated further because the odds at which she bets are not all the same. The distribution of outcomes by backing a dead cert and a hopeless long shot is very different from backing two 50-50 chances. In the first case, you almost always have one winner and one loser; in the other, you have a 25% chance of two winners, 50% of breaking even and 25% of losing both bets. The other tricky bit is adjusting our sums to account for the differences between long shots and favourites.
The way to do this is to find how many bets Alice would EXPECT to win out of her 1600. It’s time to break out some equations. I held off as long as I could. In Alice’s Great Big Spreadsheet – like all professional gamblers, she keeps meticulous records – she calculates her expected win probability as 1 divided by the odds. It’s exactly the same as the win probability – for a favourite, it’s more than a half; for an outsider it’s closer to zero. Once she’s dragged her equation down her 1600 rows, she adds it up to find she expected to win around 530.1 of the bets. She actually won 569. That’s clearly good… but is it lucky good or skilful good? We need to figure out the ‘give or take’.
The geekier type of gambler – like Alice, and like me – call the give or take margin the STANDARD DEVIATION, and it’s the square root of the VARIANCE. This is variance in a mathematical context – it’s quite precisely defined, rather than a vague “you win some, you lose some” idea. If we call the expected win probability from the previous paragraph P, the variance for any individual bet is P(1-P). It’s quite low for long shots and short favourites and relatively high when P is close to 0.5 (an evens shot). The total variance for Alice’s bets is about 339, and her standard deviation is the square root of that – 18.4.
So, armed with these numbers, how can we tell whether Alice is lucky or skilful? Now we need to look at the normal distribution. Well, I need to look at it. Alice just needs to open up her spreadsheet and type =NORMDIST(569, 530.1, 18.4) and see what comes out. The first number is the number of wins; the second number her expected wins; and the last the standard deviation. This number is how probably the results are to have come from skill rather than luck**.
In this case, the number is 0.9827 – Alice’s results can be explained 98.3% by skill and only 1.7% by luck. This is reassuring – Alice’s consistent results are much more skilful than Bob’s one-off odds-on bet.
So, if you’ve had a good run recently, it might be worth checking whether it’s the result of luck or skill using the method outlined above. I hope your numbers are way up in the nineties.
* Names have been changed to protect the fictitious.
** This method is only good when you have an expected value of more than about 30.
If you have any questions about this post please feel free to leave a comment and we shall get back to you.
“I’m trying to raise £1,000 for the DEC Haiti appeal by helping to solve people’s maths problems in exchange for donations. For more information, please click on http://www.justgiving.com/flyingcoloursmaths.” Colin is a successful bettor and math tutor. Having worked for NASA/NSF he decided to come back to the UK in 2008.